Polarization Ray Tracing and Birefringence in QUADOA® Optical CAD
QUADOA® Optical CAD models polarization in two ways: with the classical Jones and Stokes–Müller (Mueller) calculus, and with physical models of birefringent materials and birefringent thin-film coatings. This article explains why Jones matrices alone are often not enough for tilted or diverging rays, which coordinate transformation methods QUADOA® offers, which analysis tools use polarization ray tracing, and how birefringent elements are modeled with multi-sequential ray tracing. Examples with wave plates, polarizers, a birefringent crystal and an interferometer show where the approaches give different results.
Key points
- QUADOA® supports both the Jones calculus (fully polarized light, with phase) and the Stokes–Müller calculus (also partially polarized light).
- Jones and Müller matrices are exact only for collimated beams hitting the element perpendicularly. For tilted rays, one of five coordinate references (transformation methods) must be chosen, each with trade-offs.
- For unpolarized or incoherent light, use the Stokes calculus: Fresnel reflections can partially polarize the light, which affects ghost and irradiance analyses.
- All analyses, including PSF, MTF, irradiance, ghost analysis and wave optics, are based on polarization ray tracing.
- Birefringent materials are defined with ordinary and extraordinary refractive index; the ordinary and extraordinary paths are traced as two sequences with multi-sequential ray tracing.
- Birefringent materials can also be used in coating layers, for example to model thin-film polarizers and zero-order wave plates physically.
Terms used in this article
- Jones calculus
- Describes fully polarized light with a 2D complex field vector (Jones vector) and 2×2 matrices for optical elements. It keeps the phase of the field.
- Stokes–Müller calculus
- Describes light with the four Stokes parameters and 4×4 Müller (Mueller) matrices. It can represent partially polarized light, but does not contain phase information.
- Retardance
- The phase difference that an element, such as a wave plate, introduces between two orthogonal polarization components.
- Birefringence
- A material property where the refractive index depends on the polarization direction, giving an ordinary and an extraordinary ray.
- Crystal axis
- The optical axis of a birefringent crystal, which determines the ordinary and extraordinary refractive indices for a given ray direction.
- Degree of polarization
- The fraction of the light that is polarized, from 0 (unpolarized) to 1 (fully polarized).
Why are Jones and Müller matrices not always enough?
Polarization in optical software is usually simulated with one of two methods:
| Method | Light it can describe | Phase information |
|---|---|---|
| Jones calculus | Fully polarized light | Yes |
| Stokes–Müller calculus | Fully and partially polarized light | No |
An element such as a linear polarizer or wave plate is defined by a matrix in the local coordinate system of its surface. This works well when the beam hits the element perpendicularly, for example a collimated laser beam in an interferometer.
The difficulty begins when rays are tilted relative to the surface coordinate system. The electric field of the ray is a 3D vector, but the Jones and Stokes formalism is 2D. The field must therefore be transformed into the surface coordinate system before the matrix can be applied. A typical case is the pancake lens used in virtual reality headsets: polarization elements on the optical surfaces make the light reflect twice instead of passing straight through, and the rays hit these elements at many different angles.
One matrix per element is often not enough
- Angle dependence: in a wave plate, the optical path length, and therefore the retardance, depends on the angle of the ray.
- Wavelength dependence: dispersion in a birefringent material gives a different retardance for each wavelength.
- Pupil dependence: with stress-induced birefringence, the retardance varies across the pupil.
Which coordinate transformation methods does QUADOA® offer?
There is no single ideal way to transform a tilted ray into the surface coordinate system. In QUADOA®, the method is set with the Coordinate Reference (Coo. Reference) parameter, which can be chosen separately for each polarization element. The same options define the reference system of a polarized source.
| Coordinate Reference | How the ray coordinate system is derived | Behavior |
|---|---|---|
| Ideal | The surface coordinate system is rotated around k × z by the angle between ray direction k and surface normal z. This “bends” the ray into a collimated state perpendicular to the surface, applies the matrix and bends it back. | Energy is conserved; behind a linear polarizer, the polarization direction rotates with the pupil position |
| X-Axis Reference | The ray’s x axis lies in the plane spanned by the surface x axis and the ray direction | Energy is conserved; correct linear polarization for light polarized along this axis, curved polarization states for the other direction |
| Y-Axis Reference | The same with the surface y axis | As X-Axis Reference, for the y direction |
| S/P | The ray’s x axis is perpendicular to the plane of incidence (s), the y axis lies in it (p) | Natural reference for interfaces and coatings |
| Projection | The matrix is applied in the x/y plane of the surface, like projecting the field onto the surface as a shadow. The z component follows from the boundary condition and the field is renormalized so the intensity is never amplified. | Behind a linear polarizer the light is linearly polarized independent of the polarizer orientation, but parts of the field are lost, so energy is not conserved. Included for compatibility with other software; not a rigorous application of the Jones or Müller calculus |
Every method is a trade-off between energy conservation and a correct polarization state. The best choice depends on the matrix being applied and on what you analyze: polarization, intensity, or both, as in coating and ghost analyses. Projection, for example, is less suitable for illumination systems, where energy conservation matters, and should be used with caution.
How the Jones and Müller calculus is implemented in QUADOA®
Polarization elements
| Group | Elements |
|---|---|
| Static elements | Linear polarizer (transmissions Tx and Ty, so real losses and finite extinction can be modeled), circular polarizer (left or right), quarter-wave plate, half-wave plate, general retarder (retardation in waves), general Jones matrix |
| Interpolated retarders | Retardance depending on wavelength, incident angle, pupil position, or combinations of these, defined by interpolation tables |
| Müller matrix element | A general 4×4 Müller matrix |
Every element has three common parameters: Surface Action (applied on “All” interactions, only on “Transmission” or only on “Reflection”), Coo. Reference (see above) and Rz, the rotation of the element around the surface z axis. Without rotation, the main (fast) axis lies along the local surface x axis.
- Both calculi: the Jones calculus and the Stokes–Müller calculus for partially polarized light are fully supported. With a Stokes source, Jones elements are converted to their Müller equivalents automatically; with a Jones source, Müller elements are ignored.
- Any surface geometry: polarization elements can be placed on curved surfaces and aspheres, for example a retarder on a curved surface.
- Several elements per surface: for example one for reflection and one for transmission. Only one element is applied per ray–surface interaction: either chosen by its Surface Action, or selected explicitly in the surface list of a user sequence.
- Uncoated surfaces: without coating or polarization element, reflection and transmission follow the Fresnel equations, typically about 4 % reflectance at a glass–air interface, depending on angle and polarization.
Polarization of the source
| Polarization Type | Description |
|---|---|
| None | Polarization ray tracing is off; coatings are calculated for unpolarized light (average of s and p) |
| Linear | Linear polarization at an angle; 0° is horizontal |
| Circular | Left- or right-handed circular polarization |
| Jones Complex | Jones vector entered as real and imaginary parts, for any elliptical state |
| Jones Phase | Jones vector entered as amplitude and phase (in waves) of each component |
| Stokes | Stokes vector (I, Q, U, V), also for partially polarized light; phase information is not retained |
| Random | Random elliptical polarization for each ray, which behaves like unpolarized light with enough rays, with polarization ray tracing still active |
Steps
- Define the polarization state of the light in the Polarization settings of the source of the sequence, for example linear or Stokes polarization.
- Add polarization elements to a surface in the Coating tab (Polarization group), or with the right-click menu.
- Show the polarization state of the rays in the 3D view. When you rotate a wave plate, you see the polarization change immediately.
Which analysis tools use polarization ray tracing?
| Analysis | What it shows |
|---|---|
| 3D view | Polarization ellipses along the rays, useful for tracking the polarization state through complex folded systems |
| Polarization map | Polarization ellipses across the pupil |
| Ray trace report | Ray direction, position and optical path length, plus electric field, orientation and ellipticity; for partially polarized light also the Stokes vector and degree of polarization |
| Poincaré sphere | The polarization state of each ray on the Poincaré sphere, including partially polarized rays |
| PSF, MTF and wave optics | Image quality based on polarization ray tracing; the PSF differs between unpolarized and linearly polarized light |
| Irradiance and ghost irradiance | Intensity distributions, including the effect of polarization on ghosts |
For how to open the ray trace report for a single ray, see the 3D-View Editor article.
Unpolarized light: Fresnel reflections can partially polarize initially unpolarized light as it passes through a system, which can change the result of a ghost analysis. For unpolarized light with polarization ray tracing, define the source with the Stokes calculus and the vector S = (1, 0, 0, 0) instead of the Jones calculus. The Polarization Type None switches polarization ray tracing off completely.
How are birefringent materials modeled?
Instead of describing an element with a single Jones or Müller matrix, QUADOA® can model anisotropic materials physically. This avoids the transformation problem and the need for interpolation tables.
Material definition
- Each birefringent material has an ordinary and an extraordinary refractive index.
- Both can be defined with the available dispersion formulas, such as Sellmeier or SCHOTT equations, or as interpolation data of refractive index and extinction coefficient.
- In the Optical Design Editor, the material is used like any other material. Additional parameters for the crystal axis appear in the material item.
Important for tolerancing: The crystal axis is defined in the element coordinate system. Tilting a single surface, for example the first surface during tolerancing, therefore does not change the crystal axis, unlike in a classical surface-list optical model.
Tracing ordinary and extraordinary rays
With multi-sequential ray tracing, both beam paths are traced in one model:
- Set up two sequences.
- For every surface where the light enters the birefringent material, choose the ordinary path in one sequence and the extraordinary path in the other.
How are birefringent coatings modeled?
Birefringent materials can also be assigned to the layers of a coating stack. This makes it possible to model not only isotropic coatings, such as anti-reflection coatings, but also birefringent coatings, for example thin-film polarizers.
- Coating layers can be defined in different ways, and operators such as the loop operator automate the definition of repeating layers.
- Isotropic coatings are calculated with the classical transfer matrix method; birefringent coatings with an extension of this approach for anisotropic layers from the literature.
Volumetric element or coating?
| Birefringent volumetric element | Birefringent coating | |
|---|---|---|
| Sequences needed | Two (ordinary and extraordinary) | One |
| Beam displacement | Included | Ignored |
| Polarization result | Two separate polarization states; the combined state is obtained by recombining both fields in a detector plane | Combined polarization state directly |
Application examples
Birefringent crystal
A beam passes through a birefringent crystal and is analyzed with the Wave Optics Toolbox. The two sequences give a vertically and a horizontally polarized beam. Overlapping them gives the combined elliptical polarization state, and the wave-optical interferogram analysis shows the intensity and phase of the overlapping beams.
Quarter-wave plate with a diverging beam
A diverging, circularly polarized beam hits a quarter-wave plate:
| Model | Result |
|---|---|
| Ideal Jones matrix | Perfectly linear polarization and uniform illumination; the result also depends on the transformation method |
| Zero-order wave plate (single thin birefringent layer) | Ellipticity at the edge of the aperture; transmission drops for angles above about 45° |
| Multi-order wave plate (thicker birefringent layer) | Linear polarization near the optical axis, partly circular polarization towards the edge; ripples in the transmission over angle |
Linear polarizers compared
A diverging, circularly polarized beam hits three different polarizer models:
| Polarizer model | Result |
|---|---|
| Ideal Jones matrix | Perfect linear polarization; horizontal polarization fully transmitted, vertical fully absorbed |
| Absorbing polarizer (high extinction coefficient in one direction, low in the other) | Linear polarization with slight ellipticity at the edge; intensity falls off towards the side |
| Thin-film polarizer (Fresnel losses at an isotropic–anisotropic interface) | Linear polarization in the center, other states towards the edge; low fall-off up to a certain angle, then a sharp drop in intensity |
With two crossed polarizers, absorbing and thin-film polarizers give very different results. None of these effects can be reproduced with a simple Jones matrix.
Pupil-dependent birefringence
The thickness of a birefringent thin-film coating is defined by a Zernike polynomial, which makes the birefringence vary across the pupil. With a linear polarizer behind it, an image simulation shows a colorful pattern similar to stress birefringence.
Interferometer with birefringence
An interferometer is modeled with two sequences: the test arm and the reference arm. A tilted mirror produces fringes, and the wave-optical analysis agrees with the optical path difference. When an additional surface introduces birefringence, the fringes shift because of the phase change, and in some areas the contrast disappears because the polarization states of the two arms no longer match.
Best practices for polarization modeling
- Apply Jones and Müller matrices only to collimated beams. Otherwise, choose the coordinate transformation method carefully.
- Use the Stokes calculus for incoherent or unpolarized light, because Fresnel interactions can partially polarize the light.
- For birefringent elements, use multi-sequential ray tracing to follow the ordinary and extraordinary paths through the system.
- For accurate results with polarization optics such as birefringent coatings or beam splitters, prefer physical modeling over ideal matrices.
Video tutorial: polarization capabilities of QUADOA® Optical CAD
This 25-minute talk presents the polarization features of QUADOA® Optical CAD, from Jones and Stokes–Müller calculus to birefringent materials and coatings, including all examples in this article.
Video length: 25:35 min · Also on YouTube: Polarization Capabilities of QUADOA® Optical CAD
Frequently asked questions
Yes. QUADOA® fully supports the Jones calculus for fully polarized light and the Stokes–Müller calculus for partially polarized light, with predefined elements such as general Jones matrices, linear polarizers and wave plates.
A Jones matrix is defined in the surface coordinate system, while the electric field of a tilted ray is a 3D vector. The field has to be transformed into the surface system first, and every transformation method trades off energy conservation against a correct polarization state. Jones and Müller matrices are exact only for collimated beams hitting the element perpendicularly.
QUADOA® offers the coordinate references Ideal, X-Axis Reference, Y-Axis Reference, S/P and Projection, selectable per element. Projection does not conserve energy and is included only for compatibility; the other methods conserve energy but can rotate or curve the polarization state. The best choice depends on the element and on whether you analyze polarization, intensity or both.
Use the Stokes calculus with S = (1, 0, 0, 0). Fresnel reflections can partially polarize initially unpolarized light, which the Jones calculus cannot represent. This can change the result of ghost and irradiance analyses.
No. With a Jones source, Müller matrix elements are ignored. With a Stokes source, Jones elements are automatically converted to their Müller equivalents.
Define the ordinary and extraordinary refractive index with a dispersion formula such as Sellmeier or SCHOTT, or with interpolation data of refractive index and extinction coefficient. In the Optical Design Editor, the material item then shows additional parameters for the crystal axis.
With multi-sequential ray tracing. You set up two sequences and choose the ordinary path in one and the extraordinary path in the other for every surface where light enters the birefringent material.
No. The crystal axis is defined in the element coordinate system, so tilting a single surface does not change it, unlike in a classical surface-list optical model.
A birefringent coating gives the combined polarization state directly but ignores beam displacement. A volumetric element needs two sequences and includes beam displacement; the combined polarization state is obtained by recombining both fields in a detector plane.